# SageMath code for working with abstract group 972.802.
# Some of these functions may take a long time to execute (this depends on the group).


# Construction of abstract group: 
G = PermutationGroup(['(4,7)(5,8)(6,9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)', '(2,3)(5,6)(8,9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26)', '(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)', '(4,5,6)(7,9,8)(13,14,15)(16,18,17)(22,23,24)(25,27,26)', '(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)', '(1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)', '(1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)'])

# Order of the group: 
G.order()

# Exponent of the group: 
G.exponent()

# Automorphism group: 
libgap(G).AutomorphismGroup()

# Composition factors of the group: 
G.composition_series()

# Nilpotency class of the group: 
libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1

# Derived length of the group: 
libgap(G).DerivedLength()

# Determine if the group G is abelian: 
G.is_abelian()

# Determine if the group G is cyclic: 
G.is_cyclic()

# Determine if the group G is elementary abelian: 
G.is_elementary_abelian()

# Determine if the group G is nilpotent: 
G.is_nilpotent()

# Determine if the group G is perfect: 
G.is_perfect()

# Determine if the group G is a p-group: 
G.is_pgroup()

# Determine if the group G is polycyclic: 
G.is_polycyclic()

# Determine if the group G is simple: 
G.is_simple()

# Determine if the group G is solvable: 
G.is_solvable()

# Determine if the group G is supersolvable: 
G.is_supersolvable()

# Compute statistics for the group G: 
# Sage code to output the first two rows of the group statistics table
element_orders = [g.order() for g in G]
orders = sorted(list(set(element_orders)))
print("Orders:", orders)
print("Elements:", [element_orders.count(n) for n in orders], G.order())
cc_orders = [cc[0].order() for cc in G.conjugacy_classes()]
print("Conjugacy classes:", [cc_orders.count(n) for n in orders], len(cc_orders))

# List of conjugacy classes of the group: 
G.conjugacy_classes()  # Output not guaranteed to exactly match the LMFDB table

# Compute statistics about the characters of G: 
# Outputs [[d_1,c_1], [d_2,c_2], ...] where c_i is the number of irr. complex chars. of G with degree d_i
character_degrees = [c[0] for c in G.character_table()]
[[n, character_degrees.count(n)] for n in set(character_degrees)]

# Define the group with the given generators and relations: 
# This uses Sage's interface to GAP, as Sage (currently) has no native support for PC groups
GPC = gap.new('PcGroupCode(12945860361070887587763028283778178204335,972)'); a = GPC.1; b = GPC.3; c = GPC.5; d = GPC.6; e = GPC.7;

# Define the group as a permutation group: 
PermutationGroup(['(4,7)(5,8)(6,9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)', '(2,3)(5,6)(8,9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26)', '(10,11,12)(13,14,15)(16,17,18)(19,21,20)(22,24,23)(25,27,26)', '(4,5,6)(7,9,8)(13,14,15)(16,18,17)(22,23,24)(25,27,26)', '(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)', '(1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24)', '(1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)'])

# Define the group as a matrix group with coefficients in GLFp: 
MS = MatrixSpace(GF(3), 4, 4)
MatrixGroup([MS([[1, 0, 0, 0], [0, 2, 0, 0], [0, 2, 2, 1], [0, 1, 0, 1]]), MS([[0, 0, 1, 0], [0, 1, 0, 0], [2, 0, 2, 0], [1, 0, 2, 1]]), MS([[2, 0, 0, 1], [0, 1, 0, 0], [1, 0, 1, 1], [2, 0, 0, 0]]), MS([[2, 2, 0, 1], [0, 1, 0, 0], [1, 2, 1, 1], [2, 1, 0, 0]]), MS([[1, 0, 0, 0], [0, 0, 1, 1], [0, 0, 1, 0], [0, 2, 1, 2]]), MS([[0, 1, 0, 2], [0, 2, 2, 2], [2, 0, 2, 0], [1, 1, 1, 0]]), MS([[1, 0, 0, 0], [0, 2, 0, 0], [2, 0, 2, 0], [1, 0, 0, 2]])])

# Define the group from the transitive group database: 
TransitiveGroup(27, 271)
TransitiveGroup(27, 285)

# The abelianization of the group: 
G.quotient(G.commutator())

# The Schur multiplier of the group: 
G.homology(2)

# List of subgroups of the group: 
G.subgroups()

# Center of the group: 
G.center()

# Commutator subgroup of the group G: 
G.commutator()

# Frattini subgroup of the group G: 
G.frattini_subgroup()

# Fitting subgroup of the group G: 
G.fitting_subgroup()

# Socle of the group G: 
G.socle()

# Derived series of the group G: 
G.derived_series()

# Chief series of the group G: 
libgap(G).ChiefSeries()

# The lower central series of the group G: 
G.lower_central_series()

# The upper central series of the group G: 
G.upper_central_series()

# Character table: 
G.character_table()  # Output not guaranteed to exactly match the LMFDB table